Count the lines at each dot
Treat every place where lines meet, bend or end as a dot. The number of lines leaving a dot is its degree. A dot with an even number of lines is an even dot; one with an odd number is an odd dot.
- A corner the line simply turns at is still a dot (degree 2).
- A loose end where a line stops is degree 1, an odd dot.
- A point where two lines cross in an X is degree 4, an even dot.
The odd-dot rule
The mathematician Leonhard Euler presented this rule in 1735 while solving the Königsberg bridges problem. For a figure where every line is connected (no separate pieces):
| Odd dots | One line? | Start and finish |
|---|---|---|
| 0 | Yes | Start anywhere; you finish back where you started |
| 2 | Yes | Start at one odd dot, finish at the other |
| 4, 6, … | No | You must lift the pen (odd dots ÷ 2) times |
A figure with exactly 1 or 3 odd dots does not exist. Each line adds one to the degree at both of its ends, so the total of all degrees is always twice the number of lines, an even number.
The reason behind the rule is simple too. Every dot you pass through uses one line in and one line out, so it needs an even count. Only the start and the finish are allowed to be unpaired.
Choosing start and finish
- Count the degree of every dot.
- With 2 odd dots, start at one of them. Start at an even dot and you will always leave lines behind.
- With 0 odd dots, start anywhere. You will end where you began.
- With 4 or more, one stroke is impossible. How many strokes you need is covered in how many strokes an impossible figure needs.
The right start does not guarantee success; a wrong turn halfway can still trap you. See the order that keeps you from getting stuck.
Practice examples
| Figure | Odd dots | Verdict |
|---|---|---|
| Square | 0 (all four corners are 2) | Possible from anywhere |
| Square + one diagonal | 2 (diagonal ends are 3) | Start at a diagonal end |
| Square + both diagonals (X) | 4 (all corners are 3) | Impossible |
| House of Santa Claus (X square with a roof) | 2 (the two bottom corners) | Start at a bottom corner |
| Five-pointed star | 0 (all five points are 2) | Possible from anywhere |
The X square is impossible, yet putting a triangular roof on top turns the two upper corners even and makes it possible. That is the House of Santa Claus puzzle.
Checking it in the app
Show possible starting dots
With Show possible starting dots turned on in OneTrail’s settings, the two odd dots glow on boards that have a fixed start and finish. Leave it on at first and compare it with your own count.
Closed and open boards
In Endless, turning on Return to the start gives boards with 0 odd dots (start and finish at the same dot); turning it off gives boards with 2 odd dots (start and finish at different dots). Alternate between the two on the same shape and the rule soon becomes second nature.
FAQ
Do crossing points count as dots?
If you are allowed to change direction there, yes. On a dot-and-line board like OneTrail’s, just count the dots that are drawn.
There are two odd dots but I still could not finish.
Your start was right but the order went wrong, usually by using up a line you could not come back across. See the order that keeps you from getting stuck.
OneTrail: One Line Draw Puzzle
No sign-up, fully offline, and every board is checked to be solvable before you see it. Free on Android.